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IGPM462.pdf February 2017 
TITLE 
Incompressible Fluid Problems on Embedded Surfaces: Modeling and Variational Formulations 
AUTHORS 
Thomas Jankuhn, Maxim A. Olshanskii, Arnold Reusken 
ABSTRACT 
Governing equations of motion for a viscous incompressible material surface are derived from the balance laws of continuum mechanics. The surface is treated as a timedependent smooth orientable manifold of codimension one in an ambient Euclidian space. We use elementary tangential calculus to derive the governing equations in terms of exterior differential operators in Cartesian coordinates. The resulting equations can be seen as the NavierStokes equations posed on an evolving manifold. We consider a splitting of the surface NavierStokes system into coupled equations for the tangential and normal motions of the material surface. We then restrict ourselves to the case of a geometrically stationary manifold of codimension one embedded in Rn. For this case, we present new wellposedness results for the simplified surface uid model consisting of the surface Stokes equations. Finally, we propose and analyze several alternative variational formulations for these surface Stokes problem, including constrained and penalized formulations, which are convenient for Galerkin discretization methods. 
KEYWORDS 
surface PDEs, NavierStokes on surfaces 